Vending Machine: Soft Drinks A vending machine automatically...

Vending Machine: Soft Drinks A vending machine automatically pours soft drinks into cups. The amount of soft drink dispensed into a cup is normally distributed with a mean of 7.6 ounces and standard deviation of 0.4 ounce. Examine Figure $7-3$ and answer the following questions. (a) Estimate the probability that the machine will overflow an 8 -ounce cup. (b) Estimate the probability that the machine will not overflow an 8 -ounce cup. (c) The machine has just been loaded with 850 cups. How many of these do you expect will overflow when served?

Vending Machine: Soft Drinks A vending machine automatically pours soft drinks into cups. The amount of soft drink dispensed into a cup is normally distributed with a mean of 7.6 ounces and standard deviation of 0.4 ounce. Examine Figure $7-3$ and answer the following questions.
(a) Estimate the probability that the machine will overflow an 8 -ounce cup.
(b) Estimate the probability that the machine will not overflow an 8 -ounce cup.
(c) The machine has just been loaded with 850 cups. How many of these do you expect will overflow when served?

Key Concepts

Expected Value in Count Data

When dealing with a large number of independent events, the expected value is calculated by multiplying the probability of a specific outcome by the total number of trials. This concept is important in predicting outcomes in count data scenarios, such as estimating the number of occurrences of a particular event in multiple trials.

Cumulative Distribution Function

The cumulative distribution function (CDF) of a probability distribution gives the probability that a random variable is less than or equal to a specific value. For the standard normal distribution, tables or software are used to look up z-scores corresponding to given probabilities. This function is crucial in solving problems involving thresholds and specific quantiles in normally distributed data.

Complement Rule in Probability

The complement rule is a fundamental principle in probability which states that the probability of an event not occurring is equal to one minus the probability that the event does occur. This rule is useful when determining probabilities by enabling the computation of one probability when the other is easier to obtain.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric bell-shaped curve. It is defined by two parameters: the mean, which indicates the center of the distribution, and the standard deviation, which measures the spread or dispersion around the mean. It is used extensively for modeling natural phenomena and measurement errors in various fields.

Standardization and Z-Scores

Standardization is the process of converting a normally distributed random variable into a standard normal variable by subtracting the mean and dividing by the standard deviation. This yields the z-score, which indicates how many standard deviations an observation is from the mean. Z-scores allow for the comparison of values from different normal distributions and simplify the computation of probabilities.

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